This is open to interpretation. y=-5x/x^2+25 could be y=(-5x/x^2)+25 or y=-5x/(x^2+25).
The intercepts are found by setting one variable to zero and solving the equation for the other variable.
In the first interpretation, we need first to consider x close to zero, because we cannot divide by zero. -5x/x^2 is -5/x and as x tends to zero from the positive side this expression tends to negative infinity, so there is no y intercept and the y axis is an asymptote; as x tends to zero from the negative side, the expression tends to positive infinity. When y=0, 5/x=25 so the x intercept is 1/5. The graph is a hyperbola. The line y=25 is an asymptote for very large values of x (positive and negative) and there is a symmetry about this line (parallel to the x axis) and the y axis. The curve below the line and to the right of the y axis is a reflection of the the curve above the line and to the left of the y axis.
In the second interpretation, y=0 when x=0. So the origin represents both intercepts. The symmetry can be seen if we look at x=-X and x=X. y=5X/(X^2+25) and y=-5X/(X^2+25). These two values have the same magnitude but opposite signs, so there is symmetry about the x axis (y=<0), in that the shape above the x axis for negative values of x is the same as the shape below the x axis for positive values of x.